510] 
ON BICURSAL CURVES. 
185 
is the most special form to which the quadriquadric relation can be reduced by the 
linear transformations of u, v\ in fact, by mere division, the equation is made to have 
the constant term 1; the number of the coefficients of transformation is then 3+3, = 6; 
and to reduce the relation to the foregoing form, we have the six conditions, 
coeffi u = 0, coeffi v = 0, coeffi vu 2 = 0, coeff. v 2 u = 0, coeffi u 2 = 1, coeffi v 2 = 1; but in this 
form, expressing v in terms of u, the radical is *JO,, where 
H = (1 + u?) (1 + cv?) — b 2 u 2 , 
which is not more general than if we had 
H = (1 + u 2 ) (1 + cu 2 ), 
viz. there is a superfluous constant b. And we thus see how it is that the system 
of equations 
{x, y, *)=(1, u) n ( 1, v) n , 
(u, v) connected by a quadriquadric relation, contains, not 6n, but 6n+ 1 constants, one 
of these being superfluous. 
Clebsch’s transformation, above referred to, is as follows: 
Starting with the equations 
O, y, z) — (1, 6) n+k + (1, e) n+k ~ 2 
if the function © is not originally of the standard form, we may, by a linear sub 
stitution, reduce it to this form, viz. we may write 
© = 0 (1 — 0) (1 — Jc 2 0); 
and then writing 6 = sn 2 u, (sin 2 am u), we have 
V© = sn u cn u dn u 
= sn u sn' u; 
so that the formulse become 
{x, y, z) = (1, sn 2 u) n+k + (1, sn 2 u) n+k ~ 2 sn u sn' u. 
Her mite’s theorem, used in the demonstration, is that any such function of sn u 
is expressible in the form 
„ EL (m - gQ H(u- g 3 ) ct 2TO+2 fc) 
© 2 n+M (u) 
(H, © denoting here the two Jacobian functions), where 
+ a 2 ... + a 2n+2 k — 0. 
{Observe, in passing, that the equation 
0 = (1, sn 2 u) n+k + (1, sn 2 u) n+k ~ 2 sn u sn' u 
C. VIII. 
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